The Complexity of Chaos in Dynamical Systems — Epoche B2
Beyond Simple Attractors: The Complexity of Chaos in Dynamical Systems An attractor is the set of states a dissipative system ends up on once its transients have died away. Two kinds are familiar. A damped pendulum runs down to a single point of its state space and hangs still; a well-designed oscillator settles onto a closed loop and repeats forever. Both are what one would expect of a system that loses energy: it gives up its freedom until only a point or a curve is left. This essay follows the system that broke that expectation, and asks the question the textbook picture makes it hard to ask — what exactly stops a contracting flow from contracting all the way down to a point or a loop? The answer needs two separate arguments, one for why trajectories cannot escape and one for why they cannot converge, and a strange attractor exists only where both hold at once. The Lorenz system In 1963 Edward Lorenz reduced a model of a fluid layer heated from below to three ordinary differential equations [1] . A state is a point $(x,y,z)$ in three-dimensional state space: $x$ measures the intensity of the convective overturning, $y$ the temperature difference between rising and sinking fluid, and $z$ the departure of the vertical temperature profile from a straight line. All three are dimensionless. The equations are $$\begin{aligned} \frac{dx}{dt} &= \sigma(y-x), \\ \frac{dy}{dt} &= x(\rho-z)-y, \\ \frac{dz}{dt} &= xy - \beta z, \end{aligned}$$ with three positive constants: $\sigma$, the Prandtl number, the ratio of momentum diffusivity to thermal diffusivity; $\rho$, the Rayleigh number scaled so that convection begins at $\rho=1$, which measures how hard the layer is driven; and $\beta$, a geometric factor fixed by the aspect ratio of a convection roll. The only nonlinear terms in the whole system are the two products $xz$ and $xy$. Lorenz's parameters, used throughout below, are $\sigma=10$, $\rho=28$, $\beta=8/3$. The system has three equilibria. The origin, representing no convection at all, is one; the other two, representing steady overturning in either direction, sit at $$ C_{\pm} = \bigl(\pm\sqrt{\beta(\rho-1)},\ \pm\sqrt{\beta(\rho-1)},\ \rho-1\bigr) = (\pm 8.485,\ \pm 8.485,\ 27). $$ None of the three attracts. Linearising at the origin gives eigenvalues $11.83$, $-2.67$ and $-22.83$: one is positive, so the origin is a saddle. The pair $C_\pm$ are stable only while $\rho$ stays below $$ \rho_H = \frac{\sigma(\sigma+\beta+3)}{\sigma-\beta-1} = \frac{10\,(10+8/3+3)}{10-8/3-1} = 24.74, $$ and $\rho=28$ is past it — a pair of eigenvalues has crossed into the right half-plane and both steady convecting states have lost stability. So no equilibrium attracts, and numerically no periodic orbit attracts either. Something else must. Contraction: the flow squeezes volume Take a blob of initial conditions occupying a volume $V$ in state space and let every point in it evolve. The rate at which the blob's volume changes is governed by the divergence of the vector field $\mathbf{f}=(f_x,f_y,f_z)$: $$ \frac{1}{V}\frac{dV}{dt} = \nabla\cdot\mathbf{f} = \frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}. $$ For the Lorenz system the three derivatives are $-\sigma$, $-1$ and $-\beta$; the nonlinear terms contribute nothing, because $xz$ contains no $y$ and $xy$ contains no $z$. The divergence is therefore the same at every point of state space, $$ \nabla\cdot\mathbf{f} = -(\sigma+1+\beta) = -\left(10+1+\tfrac{8}{3}\right) = -13.67, $$ and because it is constant the volume equation integrates exactly, $V(t) = V(0)\,e^{-13.67\,t}$. In one unit of dimensionless time a blob shrinks to $1.2\times10^{-6}$ of its original volume; in ten units, to $10^{-59}$. Whatever the attractor is, it occupies zero volume. Contraction is not the same as boundedness It is tempting to stop there and conclude that because volume shrinks, trajectories are trapped. That does not follow, and the gap matters. Consider the linear flow $\dot{u}=u$, $\dot{w}=-2w$. Its divergence is $-1$, so volume contracts at every point, and yet $u$ grows without limit and every trajectory but one escapes to infinity. Contraction constrains a product of lengths; it says nothing about any single length. Boundedness needs its own argument. For the Lorenz system that argument uses a Lyapunov function — a quantity built from the state which can be shown to decrease. Take the one given as an exercise in Strogatz's Nonlinear Dynamics and Chaos [2] , $$ L(x,y,z) = \rho x^{2} + \sigma y^{2} + \sigma (z-2\rho)^{2}, $$ whose level sets are ellipsoids centred on $(0,0,2\rho)$, so that large $L$ means far from the origin. Differentiate along the flow and substitute the three equations: every term containing $xy$ cancels, and so does every term containing $xyz$, leaving $$ \frac{dL}{dt} = -2\sigma\Bigl[\rho x^{2} + y^{2} + \beta\left(z-\rho\right)^{2} - \beta\rho^{2}\Bigr]. $$ The bracket is positive everywhere outside the ellipsoid $\rho x^{2}+y^{2}+\beta(z-\rho)^{2} = \beta\rho^{2}$, so outside that fixed, finite region $L$ is strictly decreasing. A trajectory starting far away must fall inwards until it enters a bounded ball containing the ellipsoid, and once inside it can never get out again. That ball is a trapping region , and it, not the negative divergence, is what confines the motion. Stretching: what stops the collapse So trajectories are trapped in a bounded region and the volume they occupy tends to zero. Why is the limit not then a point or a curve? Because inside the shrinking blob one direction is being pulled apart even while the total volume falls. The rates are the Lyapunov exponents . Take two trajectories separated at $t=0$ by a small vector $\delta\mathbf{x}(0)$; along each of three principal directions the separation grows or decays exponentially, $$ |\delta\mathbf{x}(t)| \approx |\delta\mathbf{x}(0)|\,e^{\lambda t}, $$ and a three-dimensional flow has three such exponents $\lambda_1\ge\lambda_